EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5768 ISSN 1307-5543 – ejpam.com Published by New York Business Global Pythagorean Fuzzy HX-subgroups and Their Applications Areej Almuhaimeed Department of Mathematics, College of Science, Taibah University, Madinah, Saudi Arabia Abstract. In this paper, we introduce the notion of a pythagorean fuzzy HX-subgroup and a normal HX-subgroup. In addition, we prove various chracterisations for pythagorean fuzzy HX-subgroups and pythagorean normal HX-subgroups. Moreover, the notations of pythagorean fuzzy HX-subgroups homomorphisms and antihomomorphisms are introduced, and some related properties regarding the relationship between a pythagorean fuzzy set and its image are investi- gated. Characterisations of level pythagorean fuzzy HX-subgroups and normal HX-subgroups are proved. These results generalised some results regarding fuzzy HX-subgroups. 2020 Mathematics Subject Classifications: 03E72, 20N25 Key Words and Phrases: HX-groups, Pythagorean fuzzy sets, Pythagorean fuzzy homomor- phism, Pythagorean fuzzy antihomomorphism, normal HX-subgroups 1. Introduction A generalisation of the classical set, the fuzzy set notion was first presented by Zadeh in 1965 [1]. This set addressed the relationship between elements and sets and answered the question: to what extent can this object belong to a particular set, as every object x has a value η(x), where η is called a membership function η : X → [0.1]. Many ideas and abstractions have been expanded since fuzzy set theory’s inception in order to effectively handle ambiguity and uncertainty [[2], [3], [4]]. After that, researchers found that the membership function is insufficient on its own to tackle some types of situations. This motivated Atanassov [5] and [6] to introduce the idea of intuitionistic fuzzy set by associating a fuzzy set non-membership with its membership function. The non-membership η̂ and membership η in this class are satisfied: 0 ≤ η(x)+ η̂(x) ≤ 1. This set can cope with ambiguous and unclear situations more effectively than fuzzy sets since it has both a non-membership and a membership functions, see [[7], [8], [9]]. However, if the situation required η(x) + η̂(x) ≥ 1, then intuitionistic fuzzy set theory is not applicable. To find a suitable answer in these situations, Yager [10] introduced the DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5768 Email address: aamuhaimeed@taibahu.edu.sa (A. Almuhaimeed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 2 of 17 concept of pythagorean fuzzy subset. It assigns to every object x in a universe set two memberships: the membership η(x) and the non-membership η̂(x) in which 0 ≤ η(x)2 + η̂(x)2 ≤ 1. . Consequently, a pythagorean fuzzy set could be thought of as an extension of an intu- itionistic fuzzy set. Furthermore, because the condition 0 ≤ η(x)2 + η̂(x)2 ≤ 1 provides more pairs (η, η̂) than the condition 0 ≤ η(x) + η̂(x) ≤ 1, this generalisation results in a greater number of applications that can be found using pythagorean fuzzy sets than those that are solved by intuitionistic fuzzy sets. Pythagorean fuzzy sets can therefore be used to solve more issues and produce precise and efficient algorithms [[11], [12], [7]]. The concept of pythagorean fuzzy sets was applied to groups, rings and modules, see [[13], [14], [15]]. Group theory is an important branch of mathematics. It can sort numerous problems in several fields of science. The applications of group theory described in many papers [ [16], [17], [18]]. The idea of applying fuzzy settings on groups was introduced by Rosenfeld [19], who expanded on the idea of classical groups. After that, numerous efforts have been conducted to study fuzzy groups in several fuzzy environments. The notation of HX-groups was introduced in [20], by Li Hongxing. The concept has since been the subject of various studies; see [21], [22], [23], [24], [25], [23]. Several authors have merged the ideas of HX groups with the concept of fuzzy sets to get some novel findings. This study aims to establish the groundwork for a novel theory of pythagorean fuzzy HX-subgroup as it is the extension of fuzzy HX groups and intuitionistic fuzzy HX- subgroup. In this paper, we introduce the notion of a pythagorean fuzzy HX-subgroups and normal HX-subgroups. In addition, we prove various chracterisations for pythagorean fuzzy HX-subgroups and pythagorean normal HX-subgroups. Then homomorphisms of pythagorean fuzzy HX-subgroups and antihomomorphisms of pythagorean fuzzy HX- subgroups are discussed. Several related properties regarding the relationship between a pythagorean fuzzy set and its image are investigated. Moreover, pythagorean fuzzy level HX-subgroups are discussed, and characterisations of these level pythagorean fuzzy HX-subgroups and normal HX-subgroups are presented. Throughout this paper, we write PFSS to denote a pythagorean fuzzy subset, PF HX- SG to denote a pythagorean fuzzy HX-subgroup and PF HX-NSG to denote a pythagorean fuzzy HX-normal subgroup. 2. Pythagorean fuzzy HX-subgroups Recall that [26] a non empty set W ⊆ 2G − {ϕ} is called an HX group on G if W is a group with respect to algebraic operation defined by MN = {mn : m ∈ M,n ∈ N} and the identity element is denoted by e. We present the following example: A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 3 of 17 Example 1. Consider the multiplicative group G = {±1,±i}. Then the set W = {{1,−1}, {i,−i}} is an HX-group, where its identity is {1,−1}. This definition is applied to fuzzy settings in [27] and [28] and provide a definition for a fuzzy HX-SG, which defined as follows: A fuzzy set Υ is a fuzzy HX-SG of an HX-group W if, for any w01, w02 ∈ W , we have: (1) Υ (w01w02) ≥ min{Υ (w01), Υ (w02)}. (2) Υ (w−1 01 ) = Υ (w01). Now, we are able to present the main definition of PF HX-SG. Definition 1. Let G be a group, W ⊆ 2G−{ϕ} be an HX-group of G and Υ = {(w; Ῡ (w), Υ̂ (w)) : w ∈ W} be a pythagorean fuzzy subset of W . Then Υ is called a PF HX-SG of W if: (1) Ῡ 2(w01w02) ≥ min{Ῡ 2(w01), Ῡ 2(w02)}. (2) Υ̂ 2(w01w02) ≤ max{Υ̂ 2(w01), Υ̂ 2(w02)}. (3) Ῡ 2(w−1 01 ) = Ῡ 2(w01), Υ̂ 2(w−1 01 ) = Υ̂ 2(w01). Example 2. Consider the Klien 4-group G = {e, x, y, z} and the XH-group W = {E,M} = {{e, x}, {y, z}}, such that ∗ E M E E M M M E Let η be a pythagorean fuzzy sets, where η̄(e) = 0.6, η̂(e) = 0.3 η̄(x) = 0.5, η̂(x) = 0.5 η̄(y) = 0.4, η̂(y) = 0.3 η̄(z) = 0.3, η̂(z) = 0.5 Let Ῡ (n) = max{η̄(n) : n ∈ N ⊆ W} and Υ̂ (n) = min{η̂(n) : n ∈ N ⊆ W}. Thus Ῡ (E) = max{η̄(e), η̄(x)} = 0.6 Υ̂ (E) = min{η̂(e), η̂(x)} = 0.3 Ῡ (M) = max{η̄(y), η̄(z)} = 0.4 Υ̂ (M) = min{η̂(y), η̂(z)} = 0.3 Thus Υ is a PF HX-SG. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 4 of 17 Now, we inroduce the following proposition which will be used in later results. Proposition 1. Let Υ be a PF HX-SG of an HX-group W . Then Ῡ 2(e) ≥ Ῡ 2(w01), Υ̂ 2(e) ≤ Υ̂ 2(w01) for all w01 ∈ W Proof. Ῡ 2(e) = Ῡ 2(w01w −1 01 ) ≥ min{Ῡ 2(w01), Ῡ 2(w−1 01 )} = min{Ῡ 2(w01), Ῡ 2(w01)} = Ῡ 2(w01) Υ̂ 2(e) = Υ̂ 2(w01w −1 01 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w−1 01 )} = max{Υ̂ 2(w01), Υ̂ 2(w01)} = Υ̂ 2(w01). Using the above proposition, we are able to provide characterisation theorem of HX- SG. Theorem 1. Let W be an HX-group and Υ be a PF subset of W . Then Υ is a PF HX-SG of W if and only if Ῡ 2(w01w −1 02 ) ≥ min{Ῡ 2(w01), Ῡ 2(w02)}, Υ̂ 2(w01w −1 02 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w02)}. (1) Proof. If Υ is a PF HX-SG of W , then Ῡ 2(w01w −1 02 ) ≥ min{Ῡ 2(w01), Ῡ 2(w−1 02 )} = min{Ῡ 2(w01), Ῡ 2(w02)} Also,Υ̂ 2(w01w −1 02 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w−1 02 )} = max{Υ̂ 2(w01), Υ̂ 2(w02)}. If (1) holds, then Ῡ 2(w−1 01 ) = Ῡ 2(ew−1 01 ) ≥ min{Ῡ 2(e), Ῡ 2(w01)} = Ῡ 2(w01). On the other hand, Ῡ 2(w01) = Ῡ 2(ew01) ≥ min{Ῡ 2(e), Ῡ 2(w−1 01 )} = Ῡ 2(w−1 01 ). Thus Ῡ 2(w−1 01 ) = Ῡ 2(w01). In addition, Ῡ 2(w01w02) = Ῡ 2(w01(w −1 02 ) −1 ) ≥ min{Ῡ 2(w01), Ῡ 2(w−1 02 )} = min{Ῡ 2(w01), Ῡ 2(w02)}. Similarly, Υ̂ 2(w−1 01 ) = Υ̂ 2(ew−1 01 ) ≤ max{Υ̂ 2(e), Υ̂ 2(w01)} = Υ̂ 2(w01) and Υ̂ 2(w01) = Υ̂ 2(ew01) ≤ max{Υ̂ 2(e), Υ̂ 2(w−1 01 )} = Υ̂ 2(w−1 01 ). Thus Υ̂ 2(w−1 01 ) = Υ̂ 2(w01). In addition, Υ̂ 2(w01w02) = Υ 2(w01(w −1 02 ) −1 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w−1 02 )} = max{Υ̂ 2(w01), Υ̂ 2(w02)} Hence Υ is a PF HX-SG of W . A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 5 of 17 Proposition 2. Let W be an HX-group and Υi be PF HX-SGs of W . Then ⋂ i Υi is a PF HX-SG of W . Proof. Clear. We define two well known pytharorean fuzzy subsets: Definition 2. Let W be an HX-group and P be a PFSS of W . Then (1) P ⋆ = Ῡ ⋆ P ∩ Υ̂ ⋆ P , where Ῡ ⋆ P = {w ∈ W : Ῡ 2 P (w) > 0} Υ̂ ⋆ P = {w ∈ W : Υ̂ 2(w) < 1} (2) P⋆ = Ῡ⋆P ∩ Υ̂⋆P , where Ῡ⋆P = {w ∈ W : Ῡ 2 P (w) = 1} Υ̂⋆P = {w ∈ W : Υ̂ 2(w) = 0} Now, we prove that the above PFSSs are subgroups of G in the case that ΥP is a PF HX-SG. Theorem 2. If ΥP is a PF HX-SG of G, then P ⋆ is a subgroup of G. Proof. Suppose that w01, w02 ∈ P ⋆. Then Ῡ 2(w01) > 0, Ῡ 2(w02) > 0 and Υ̂ 2(w01) < 1, Υ̂ 2(w02) < 1. By hypothesis, Ῡ 2(w01w −1 02 ) ≥ min{Ῡ 2(w01), Ῡ 2(w02)} > 0. Also, Υ̂ 2(w01w −1 02 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w02)} < 1. Hence w01w −1 02 ∈ P ⋆ and therefore, P ⋆ is a subgroup of G. Theorem 3. If ΥP is a PF HX-SG of G, then P⋆ is a subgroup of G. Proof. Assume that w01, w02 ∈ P ⋆. Then Ῡ 2(w01) = 1, Ῡ 2(w02) = 1 and Υ̂ 2(w01) = 0, Υ̂ 2(w02) = 0. By hypothesis, Ῡ 2(w01w −1 02 ) ≥ min{Ῡ 2(w01), Ῡ 2(w02)} = 1. Also, Υ̂ 2(w01w −1 02 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w02)} = 0. Hence w01w −1 02 ∈ P⋆ and therefore, P⋆ is a subgroup of G. A (ζ, δ)-level pythagorean fuzzy subset can be defined as follows: Definition 3. Let Υ be a PFSS of W . Then Υ(ζ,δ) = {w ∈ W : Ῡ 2(w) ≥ ζ and Υ̂ 2(w) ≤ δ}, where ζ, δ ∈ [0, 1]. An essential question arises: is there a relationship between these level PFSSs Υ(ζ,δ) and a PFSS Υ? We present the following theorem to answer this question. Theorem 4. Let Υ be a PFSS of W . Then Υ(ζ,δ) is an HX-SG, for all ζ, δ ∈ [0, 1], if and only if Υ is a PF HX-SG. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 6 of 17 Proof. Suppose that Υ(ζ,δ) is an HX-SG. Assume that there exists w01, w02 ∈ W such that Ῡ 2(w01w −1 02 ) < min{Ῡ 2(w01), Ῡ 2(w02)}. Then Ῡ 2(w01w −1 02 ) < ζ, for some ζ ∈ [0, 1] and this contradicts that Υ(ζ,δ) is an HX-SG. Thus Ῡ 2(w01w −1 02 ) ≥ min{Ῡ 2(w01), Ῡ 2(w02)}. Now, let Υ̂ 2(w01w −1 02 ) > max{Υ̂ 2(w01), Υ̂ 2(w02)}. This implies that Υ̂ 2(w01w −1 02 ) > δ, for some δ ∈ [0, 1] and this contradicts that Υ(ζ,δ) is an HX-SG. This menas that Υ̂ 2(w01w −1 02 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w02)}. Hence Υ is a PF HX-SG. Conversely, suppose that Υ is a PF HX-SG. Let w01, w02 ∈ Υ(ζ,δ), for some (ζ, δ). Then Ῡ 2(w01) ≥ ζ, Ῡ 2(w02) ≥ ζ, Υ̂ 2(w01) ≤ δ and Υ̂ 2(w02) ≤ δ. That Υ is a PF HX-SG implies that Ῡ 2(w01w −1 02 ) ≥ min{Ῡ 2(w01), Ῡ 2(w02)} ≥ ζ and Υ̂ 2(w01w −1 02 ) ≤ max{Υ̂ 2(w01), Υ̂ 2(w02)} ≤ δ. Hence w01w −1 02 ∈ Υ(ζ,δ) and therefore, Υ(ζ,δ) is an HX-SG. 3. Pythagorean fuzzy HX-normal subgroups In this section, we study PF HX-NSGs. We first introduce the definition of left cosets: Definition 4. Let Υ be an HX-SG of W . Then a left coset of Υ in W is defined by w01Υ (w02) = Υ (w−1 01 w02) for all w01, w02 ∈ W , that is w01Ῡ 2(w02) = Ῡ 2(w−1 01 w02) and w01Υ̂ 2(w02) = Υ̂ 2(w−1 01 w02) for all w01, w02 ∈ W . In order to explain the above definition, we present the following example: Example 3. Consider the group (Z∗ 7, ·7) and the XH-group W = {E,M,N} = {{1, 6}, {2, 5}, {3, 4}}, such that ∗ E M N E E M N M M N E N N E M Let η be a pythagorean fuzzy set, where η̄(1) = 0.6, η̂(1) = 0.7 η̄(2) = 0.5, η̂(2) = 0.6 η̄(3) = 0.3, η̂(3) = 0.5 η̄(4) = 0.5, η̂(4) = 0.4 η̄(5) = 0.4, η̂(5) = 0.4 η̄(6) = 0.5, η̂(6) = 0.5 Let Ῡ (n) = max{η̄(n) : n ∈ N ⊆ W} and Υ̂ (n) = min{η̂(n) : n ∈ N ⊆ W}. Thus Ῡ (E) = max{η̄(1), η̄(6)} = 0.6 A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 7 of 17 Υ̂ (E) = min{η̂(1), η̂(6)} = 0.5 Ῡ (M) = max{η̄(2), η̄(5)} = 0.5 Υ̂ (M) = min{η̂(2), η̂(5)} = 0.4 Ῡ (N) = max{η̄(3), η̄(4)} = 0.5 Υ̂ (N) = min{η̂(3), η̂(4)} = 0.4 We find the left coset MΥ (N). By definition, MΥ (N) = Υ (M−1N). That is: MῩ 2(N) = Ῡ 2(M−1N) = Ῡ 2(NN) = Ῡ 2(M) = 0.36 MΥ̂ 2(N) = Υ̂ 2(M−1N) = Υ̂ 2(NN) = Υ̂ 2(M) = 0.16. Similarly, we can compute any left coset. Now, we are ready to define a PF HX-NSG. Definition 5. Let G be a group, W ⊆ 2G−{ϕ} be an HX-group of G and Υ = {(w; Ῡ (w), Υ̂ (w)) : w ∈ W} be a PF HX-SG of W . Then Υ is called a PF HX-NSG of W if: Ῡ 2(w01w02) = Ῡ 2(w02w01) and Υ̂ 2(w01w02) = Υ̂ 2(w02w01). Alternatively, Υ is a PF HX-NSG of W if w01Υ (w02) = Υ (w02)w01 for all w01, w02 ∈ W . Example 4. Consider W , η and Υ as in example 3. Then: Ῡ 2(EM) = Ῡ 2(ME) = 0.25 Υ̂ 2(EM) = Υ̂ 2(ME) = 0.16 Ῡ 2(EN) = Ῡ 2(NE) = 0.25 Υ̂ 2(EN) = Υ̂ 2(NE) = 0.16 Ῡ 2(NM) = Ῡ 2(MN) = 0.36 Υ̂ 2(NM) = Υ̂ 2(MN) = 0.25 Thus Υ is a PF HXN-SG. Turning to PF HX-NSGs, we can say more about the properties of them. The following theorems described these properties: Theorem 5. Let W be an HX-group and Υ be a PF HX-SG of W . Then Υ is a PF HX-NSG of W if and only if Ῡ 2(w−1 01 w02w01) = Ῡ 2(w02), Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w02). (2) Proof. If Υ is a PF HX-NSG of W , then Ῡ 2(w−1 01 w02w01) = Ῡ 2((w−1 01 w02)w01) A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 8 of 17 = Ῡ 2((w02w −1 01 )w01) = Ῡ 2(w02(w −1 01 w01)) = Ῡ 2(w02) and Υ̂ 2(w−1 01 w02w01) = Υ̂ 2((w−1 01 w02)w01) = Υ̂ 2((w02w −1 01 )w01) = Υ̂ 2(w02(w −1 01 w01)) = Υ̂ 2(w02) Conversely, if (2) holds, then Ῡ 2(w01w02) = w−1 01 Ῡ 2(w02) = w−1 01 Ῡ 2(w−1 01 w02w01) = Ῡ 2(w01w −1 01 w02w01) = Ῡ 2(w02w01) and Υ̂ 2(w01w02) = w−1 01 Υ̂ 2(w02) = w−1 01 Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w01w −1 01 w02w01) = Υ̂ 2(w02w01) Hence Υ is a PF HX-NSG. Proposition 3. Let W be an HX-group and Υi be PF HX-NSGs of W . Then ⋂ i Υi is a PF HX-NSG of W . Proof. Clear. The following two theorems are the analogue of theorem 2 and theorem 3: Theorem 6. If ΥP is a PF HX-NSG of G, then P ⋆ is a normal subgroup of G. Proof. Since ΥP is a PF HX-NSG of G, then ΥP is a PF HX-SG of G, thus, by theorem 2, P ⋆ is a subgroup of G. Now, suppose that w01, w02 ∈ P ⋆. Then Ῡ 2(w01) > 0, Ῡ 2(w02) > 0 and Υ̂ 2(w01) < 1, Υ̂ 2(w02) < 1. That ΥP is a PF HX-NSG implies that Ῡ 2(w−1 01 w02w01) = Ῡ 2(w02)} > 0 and Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w02)} < 1. Hence w−1 01 w02w01 ∈ P ⋆ and therefore, P ⋆ is a normal subgroup of G. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 9 of 17 Theorem 7. If ΥP is a PF HX-NSG of G, then P⋆ is a normal subgroup of G. Proof. Since ΥP is a PF HX-NSG of G, then ΥP is a PF HX-SG of G. Thus, by theorem 3, P⋆ is a subgroup of G. Now, let w01, w02 ∈ P⋆. Then Ῡ 2(w01) = 1, Ῡ 2(w02) = 1 and Υ̂ 2(w01) = 0, Υ̂ 2(w02) = 0. That ΥP is a PF HX-NSG implies that Ῡ 2(w−1 01 w02w01) = Ῡ 2(w02)} = 1 and Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w02)} = 0. Hence w−1 01 w02w01 ∈ P⋆ and therefore, P ⋆ is a normal subgroup of G. For (ζ, δ)-level PFSSs, we prove the following theorem: Theorem 8. Let Υ be a PFSS of W . Then Υ(ζ,δ) is a PF HX-NSG if and only if Υ is a PF HX-NSG. Proof. Suppose that Υ(ζ,δ) is a PF HX-NSG. Then Υ(ζ,δ) is a PF HX-SG and hence by theorem 4, Υ is a PF HX-SG. Assume that there exists w01, w02 ∈ W such that Ῡ 2(w−1 01 w02w01) < Ῡ 2(w02) = ζ, for some ζ ∈ [0, 1]. Then Ῡ 2(w−1 01 w02w01 < ζ and this contradicts that Υ(ζ,δ) is a PF HX-NSG. In the case that Ῡ 2(w02) < Ῡ 2(w−1 01 w02w01), then Ῡ 2(w02) < Ῡ 2(w−1 01 w02w01) = ζ, for some ζ ∈ [0, 1] and this contradicts that Υ(ζ,δ) is a PF HX-NSG. Thus Ῡ 2(w−1 01 w02w01) = Ῡ 2(w02). Now, let Υ̂ 2(w−1 01 w02w01) > Υ̂ 2(w02) = δ, for some δ ∈ [0, 1]. Then Υ̂ 2(w−1 01 w02w01) > δ and this contradicts that Υ(ζ,δ) is a PF HX- NSG. In the case that Υ̂ 2(w02) > Υ̂ 2(w−1 01 w02w01), then Υ̂ 2(w02) > Υ̂ 2(w−1 01 w02w01) = ζ, for some ζ ∈ [0, 1] and this contradicts that Υ(ζ,δ) is a PF HX-NSG. Thus Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w02). Thus Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w02). Therefore, Υ is a PF HX-NSG. Conversely, assume that Υ is a PF HX-NSG. Let w01, w02 ∈ Υ(ζ,δ), for some (ζ, δ). Then Ῡ 2(w01) ≥ ζ, Ῡ 2(w02) ≥ ζ, Υ̂ 2(w01) ≤ δ and Υ̂ 2(w02) ≤ δ. That Υ is a PF HX-NSG implies that Ῡ 2(w−1 01 w02w01) = Ῡ 2(w02)} ≥ ζ and Υ̂ 2(w−1 01 w02w01) = Υ̂ 2(w02) ≤ δ. Hence w−1 01 w02w01 ∈ Υ(ζ,δ) and therefore, Υ(ζ,δ) is a PF HX-NSG. 4. Pythagorean fuzzy homomorphism of HX-subgroups Let P, S be two HX-groups, L a PF HX-SG of P and N a PF-HX-SG of S. Consider a homomorphism τ : P −→ S For s ∈ S, we define: ητ(L)(s) = { max{ῩL(p) : s = τ(p)} if s ∈ Im(τ) 0 otherwise and η̂τ(L)(s) = { min{Υ̂L(p) : s = τ(p)} if s ∈ Im(τ) 1 otherwise After presenting the definition of PF homomorphism, we are now ready to provide a relationship between a PFSS and its image. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 10 of 17 Theorem 9. Let τ : P −→ S be a homomorphism of HX-groups. If Υ is a PF-HX-SG of P , then τ(Υ ) is a PF-HX-SG of S. Proof. Suppose that τ(Υ ) = {(τ(p), η, η̂) : τ(p) ∈ S}. Let τ(p01), τ(p01) ∈ S. Then η2(τ(p01)(τ(p02)) −1) = η2(τ(p01)τ(p −1 02 )) = η2(τ(p01p −1 02 ) ≥ Ῡ 2(p01p −1 02 ) ≥ min{Ῡ 2(p01, Ῡ 2(p02)} = min{η2(τ(p01)), η2(τ(p02))} Also, η̂2(τ(p01)(τ(p02)) −1) = η̂2(τ(p01)τ(p −1 02 )) = η̂2(τ(p01p −1 02 ) ≤ Υ̂ 2(p01p −1 02 ) ≤ max{Υ̂ 2(p01, Υ̂ 2(p02)} = max{η̂2(τ(p01)), η̂2(τ(p02))} Hence τ(Υ ) is a PF-HX-SG of S. Theorem 10. Let τ : P −→ S be a homomorphism of HX-groups. If Υ is a PF-HX-NSG of P , then τ(Υ ) is a PF HX-NSG of S. Proof. Suppose that τ(Υ ) = {(τ(p), η, η̂) : τ(p) ∈ S}. Since Υ is a PF-HX-NSG of P , then it is a PF-HX-SG and, by the previous theorem, τ(Υ ) is a PF-HX-SG of S. We need to prove that it is normal. Let τ(p01), τ(p01) ∈ S. Then η2(τ(p01)τ(p02)) = (τ(p01)) −1η2(τ(p02)) = (τ(p01)) −1Ῡ 2((p02) = (τ(p01)) −1Ῡ 2((p01) −1p02p01) ≤ (τ(p01)) −1η2(τ((p01) −1p02p01)) = η2(τ(p01)τ((p01) −1p02p01)) = η2(τ(p01(p01) −1p02p01)) = η2(τ(p02p01)) = η2(τ(p02τ(p01))) Thus η2(τ(p01)τ(p02)) ≤ η2(τ(p02τ(p01))). On the other hand, η2(τ(p02)τ(p01)) = (τ(p02)) −1η2(τ(p01) A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 11 of 17 = (τ(p02)) −1Ῡ 2((p01) = (τ(p02)) −1Ῡ 2((p02) −1p01p02) ≤ (τ(p02)) −1η2(τ((p02) −1p01p02)) = η2(τ(p02)τ((p02) −1p01p02)) = η2(τ(p02(p02) −1p01p02)) = η2(τ(p01p02)) = η2(τ(p01τ(p02))). Then η2(τ(p02)τ(p01)) ≤ η2(τ(p01τ(p02))). Hence η2(τ(p01τ(p02))) = η2(τ(p02τ(p01))). In addition, η̂2(τ(p01)τ(p02)) = (τ(p01)) −1η̂2(τ(p02)) = (τ(p01)) −1Υ̂ 2((p02) = (τ(p01)) −1Υ̂ 2((p01) −1p02p01) ≥ (τ(p01)) −1η̂2(τ((p01) −1p02p01)) = η̂2(τ(p01)[]τ((p01) −1p02p01)) = η̂2(τ(p01(p01) −1p02p01)) = η̂2(τ(p02p01)) = η̂2(τ(p02τ(p01))) This implies that η̂2(τ(p01)τ(p02)) ≥ η̂2(τ(p02τ(p01))). On the other hand, η̂2(τ(p02)τ(p01)) = (τ(p02)) −1η̂2(τ(p01) = (τ(p02)) −1Υ̂ 2((p01) = (τ(p02)) −1Υ̂ 2((p02) −1p01p02) ≥ (τ(p02)) −1η̂2(τ((p02) −1p01p02)) = η̂2(τ(p02)τ((p02) −1p01p02)) = η̂2(τ(p02(p02) −1p01p02)) = η̂2(τ(p01p02)) = η̂2(τ(p01τ(p02))) Then η̂2(τ(p02)τ(p01)) ≥ η̂2(τ(p01τ(p02))). Hence η̂2(τ(p01τ(p02))) = η̂2(τ(p02τ(p01))). Therefore, τ(Υ ) is a PF-NHX-SG of S. Theorem 11. Let τ : P −→ S be an antihomomorphism of HX-groups. If Υ is a PF-HX- SG of P , then τ(Υ ) is a PF-HX-SG of S. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 12 of 17 Proof. Suppose that τ(Υ ) = {(τ(p), η, η̂) : τ(p) ∈ S}. Let τ(p01), τ(p01) ∈ S. Then η2(τ(p01)(τ(p02)) −1) = η2(τ(p01)τ(p −1 02 )) = η2(τ(p−1 02 p01) ≥ Ῡ 2(p−1 02 p01) ≥ min{Ῡ 2(p02) −1, Ῡ 2(p01)} = min{Ῡ 2(p01), Ῡ 2(p02)} = min{η2(τ(p01)), η2(τ(p02))} Also, η̂2(τ(p01)(τ(p02)) −1) = η̂2(τ(p01)τ(p −1 02 )) = η̂2(τ(p−1 02 p01) ≤ Υ̂ 2(p−1 02 p01) ≤ max{Υ̂ 2(p−1 02 , Υ̂ 2(p01)} = max{Υ̂ 2(p01, Υ̂ 2(p02)} = max{η̂2(τ(p01)), η̂2(τ(p02))}. Hence τ(Υ ) is a PF-HX-SG of S. Theorem 12. Let τ : P −→ S be an antihomomorphism of HX-groups. If Υ is a PF-HX- NSG of P , then τ(Υ ) is a PF-NHX-SG of S. Proof. Suppose that τ(Υ ) = {(τ(p), η, η̂) : τ(p) ∈ S}. Since Υ is a PF-HX-NSG of P , then it is a PF-HX-SG and, by the previous theorem, τ(Υ ) is a PF-HX-SG of S. We need to prove that it is normal. Let τ(p01), τ(p01) ∈ S. Then η2(τ(p01)τ(p02)) = (τ(p01)) −1η2(τ(p02)) = (τ(p01)) −1Ῡ 2((p02) = (τ(p01)) −1Ῡ 2(p01p02(p01) −1) ≤ (τ(p01)) −1η2(τ(p01p02(p01) −1)) = η2(τ(p01)τ(p01p02(p01) −1)) = η2(τ(p01p02(p01) −1p01)) = η2(τ(p01p02)) = η2(τ(p02)τ(p01))) Then η2(τ(p01τ(p02))) ≤ η2(τ(p02τ(p01))). On the other hand, η2(τ(p02)τ(p01)) = (τ(p02)) −1η2(τ(p01)) A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 13 of 17 = (τ(p02)) −1Ῡ 2((p01) = (τ(p02)) −1Ῡ 2(p02p01(p02) −1) ≤ (τ(p02)) −1η2(τ(p02p01(p02) −1)) = η2(τ(p02)τ(p02p01(p02) −1)) = η2(τ(p02p01(p02) −1p02)) = η2(τ(p02p01)) = η2(τ(p01)τ(p02))) This impliea that η2(τ(p02τ(p02))) ≤ η2(τ(p01τ(p02))). Thus η 2(τ(p01τ(p02))) = η2(τ(p02τ(p01))). In addition, η̂2(τ(p01)τ(p02)) = (τ(p01)) −1η̂2(τ(p02)) = (τ(p01)) −1Υ̂ 2((p02) = (τ(p01)) −1Υ̂ 2(p01p02(p01) −1) ≤ (τ(p01)) −1η̂2(τ(p01p02(p01) −1)) = η̂2(τ(p01)τ(p01p02(p01) −1)) = η̂2(τ(p01p02(p01) −1p01)) = η̂2(τ(p01p02)) = η̂2(τ(p02)τ(p01))) On the other hand, η̂2(τ(p02)τ(p01)) = (τ(p02)) −1η̂2(τ(p01)) = (τ(p02)) −1Υ̂ 2((p01) = (τ(p02)) −1Υ̂ 2(p02p01(p02) −1) ≤ (τ(p02)) −1η̂2(τ(p02p01(p02) −1)) = η̂2(τ(p02)τ(p02p01(p02) −1)) = η̂2(τ(p02p01(p02) −1p02)) = η̂2(τ(p02p01)) = η̂2(τ(p01)τ(p02))) Thus η̂2(τ(p01τ(p02))) = η̂2(τ(p02τ(p01))). Therefore, τ(Υ ) is a PF-NHX-SG of S. Theorem 13. Let τ : P −→ S be a homomorphism of HX-groups and let Υ 2 be a PFSS of W . If Υ(ζ,δ) is an HX-SG, then τ(Υ(ζ,δ)) is an HX-SG. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 14 of 17 Proof. Suppose that τ(Υ(ζ,δ)) = {(τ(p), η2, η̂2) : p ∈ Υ(ζ,δ)}. Let τ(p01), τ(p02) ∈ τ(Υ(ζ,δ)). Then η2(τ(p01)) ≥ ζ, η2(τ(p02)) ≥ ζ, η̂2(τ(p01)) ≤ δ and η̂2(τ(p02)) ≤ δ. Then η2(τ(p01)τ(p02) −1) = η2(τ(p01)τ(p −1 02 )) = η2(τ(p01p −1 02 )) ≥ Ῡ 2(p01p −1 02 ) Since p01, p02 ∈ Υ(ζ,δ) and Υ(ζ,δ) is an HX-SG, then p01p −1 02 ∈ Υ(ζ,δ) which implies that Ῡ 2(p01p −1 02 ) ≥ ζ. Thus η2(τ(p01)τ(p02) −1) ≥ ζ. In addition, η̂2(τ(p01)τ(p02) −1) = η̂2(τ(p01)τ(p −1 02 )) = η̂2(τ(p01p −1 02 )) ≤ Υ̂ 2(p01p −1 02 ) Since p01, p02 ∈ Υ(ζ,δ) and Υ(ζ,δ) is an HX-SG, it implies that Υ̂ 2(p01p −1 02 ) ≤ δ. Thus η̂2(τ(p01)τ(p02) −1) ≤ δ. Therefore, τ(Υ(ζ,δ)) is an HX-SG. Theorem 14. Let τ : P −→ S be a homomorphism of HX-groups and let Υ 2 be a PFSS of W . If Υ(ζ,δ) is an HX-NSG, then τ(Υ(ζ,δ)) is an HX-NSG. Proof. Since Υ(ζ,δ) is an HX-NSG, then it is an HX-SG and, by the previous theo- rem, τ(Υ(ζ,δ)) is an HX-SG. We need to prove that it is normal. Suppose that τ(Υ(ζ,δ)) = {(τ(p), η2, η̂2) : p ∈ Υ(ζ,δ)}. Let τ(p01), τ(p02) ∈ τ(Υ(ζ,δ)). Then η2(τ(p01)) ≥ ζ, η2(τ(p02)) ≥ ζ, η̂2(τ(p01)) ≤ δ and η̂2(τ(p02)) ≤ δ. Then η2(τ(p01) −1τ(p02)τ(p01)) = η2(τ(p−1 01 )τ(p02)τ(p01)) = η2(τ(p−1 01 p02p01)) ≥ Ῡ 2(p−1 01 p02p01) Since p01, p02 ∈ Υ(ζ,δ) and Υ(ζ,δ) is an HX-NSG, then p−1 01 p02p01 ∈ Υ(ζ,δ). Thus Ῡ 2(p−1 01 p02p01) ≥ ζ. Hence η2(τ(p01) −1τ(p02)τ(p01)) ≥ ζ. Moreover, η̂2(τ(p01) −1τ(p02)τ(p01)) = η̂2(τ(p−1 01 )τ(p02)τ(p01)) = η̂2(τ(p−1 01 p02p01)) ≤ Υ̂ 2(p−1 01 p02p01) Since p01, p02 ∈ Υ(ζ,δ) and Υ(ζ,δ) is an HX-NSG, then p−1 01 p02p01 ∈ Υ(ζ,δ). Thus Υ̂ 2(p−1 01 p02p01) ≤ δ. Hence η̂2(τ(p01) −1τ(p02)τ(p01)) ≤ δ. Therefore, τ(Υ(ζ,δ)) is an HX-NSG. Theorem 15. Let τ : P −→ S be an antihomomorphism of HX-groups and let Υ 2 be a PFSS of W . If Υ(ζ,δ) is an HX-SG, then τ(Υ(ζ,δ)) is an HX-SG. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 15 of 17 Proof. Suppose that τ(Υ(ζ,δ)) = {(τ(p), η2, η̂2) : p ∈ Υ(ζ,δ)}. Let τ(p01), τ(p02) ∈ τ(Υ(ζ,δ)). Then η2(τ(p01)) ≥ ζ, η2(τ(p02)) ≥ ζ, η̂2(τ(p01)) ≤ δ and η̂2(τ(p02)) ≤ δ. Then η2(τ(p01)τ(p02) −1) = η2(τ(p01)τ(p −1 02 )) = η2(τ(p−1 02 p01)) ≥ Ῡ 2(p−1 02 p01) Since p01, p02 ∈ Υ(ζ,δ) and Υ(ζ,δ) is an HX-SG, then p−1 01 , p −1 02 ∈ Υ(ζ,δ) and so p−1 02 (p −1 01 ) −1 ∈ Υ(ζ,δ). Now, Ῡ 2(p−1 02 p01) = Ῡ 2(p−1 02 (p −1 01 ) −1) ≥ ζ. Thus η2(τ(p01)τ(p02) −1) ≥ ζ. In addition, η̂2(τ(p01)τ(p02) −1) = η̂2(τ(p01)τ(p −1 02 )) = η̂2(τ(p−1 02 p01)) ≤ Υ̂ 2(p−1 02 p01) Since p−1 02 (p −1 01 ) −1 ∈ Υ(ζ,δ), it implies that Υ̂ 2(p−1 02 p01) = Υ̂ 2(p−1 02 (p −1 01 ) −1) ≤ δ. Thus η̂2(τ(p01)τ(p02) −1) ≤ δ. Therefore, τ(Υ(ζ,δ)) is an HX-SG. Theorem 16. Let τ : P −→ S be a homomorphism of HX-groups and let Υ 2 be a PFSS of W . If Υ(ζ,δ) is an HX-NSG, then τ(Υ(ζ,δ)) is an HX-NSG. Proof. Since Υ(ζ,δ) is an HX-NSG, then it is an HX-SG and, by the previous theo- rem, τ(Υ(ζ,δ)) is an HX-SG. We need to prove that it is normal. Suppose that τ(Υ(ζ,δ)) = {(τ(p), η2, η̂2) : p ∈ Υ(ζ,δ)}. Let τ(p01), τ(p02) ∈ τ(Υ(ζ,δ)). Then η2(τ(p01)) ≥ ζ, η2(τ(p02)) ≥ ζ, η̂2(τ(p01)) ≤ δ and η̂2(τ(p02)) ≤ δ. Then η2(τ(p01) −1τ(p02)τ(p01)) = η2(τ(p−1 01 )τ(p02)τ(p01)) = η2(τ(p02p −1 01 )τ(p01)) = η2(τ(p01p02p −1 01 )) ≥ Ῡ 2(p01p02p −1 01 ). Since p01, p02 ∈ Υ(ζ,δ) and Υ(ζ,δ) is an HX-NSG, then p−1 01 , p02 ∈ Υ(ζ,δ). This implies that p01p02p −1 01 = (p−1 01 ) −1p02p −1 01 ∈ Υ(ζ,δ). Thus Ῡ 2(p01p02p −1 01 ) ≥ ζ. Hence η2(τ(p01) −1τ(p02)τ(p01)) ≥ ζ. Moreover, η̂2(τ(p01) −1τ(p02)τ(p01)) = η̂2(τ(p−1 01 )τ(p02)τ(p01)) = η̂2(τ(p02p −1 01 )τ(p01)) = η̂2(τ(p01p02p −1 01 )) ≤ Υ̂ 2(p01p02p −1 01 ). Since p−1 01 , p02 ∈ Υ(ζ,δ), then p01p02p −1 01 = (p−1 01 ) −1p02p −1 01 ∈ Υ(ζ,δ). Thus Υ̂ 2(p01p02p −1 01 ) ≤ δ. Hence η̂2(τ(p01) −1τ(p02)τ(p01)) ≤ δ. A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 16 of 17 5. Conclusion This article introduced the novel concept of a pythagorean fuzzy HX-subgroup and a normal HX-subgroup. This study aims to establish the groundwork for a novel theory of pythagorean fuzzy HX-subgroups as it is the extension of fuzzy HX groups and intuitionis- tic fuzzy HX-subgroup. Various chracterisations for pythagorean fuzzy HX-subgroups and pythagorean normal HX-subgroups are proved. Moreover, the notations of pythagorean fuzzy HX-subgroups homomorphisms and antihomomorphisms are initiated, and some re- lated properties regarding the relationship between a pythagorean fuzzy set and its image are investigated. Characterisations of level pythagorean fuzzy HX-subgroups and normal HX-subgroups are presented. In future work, this study can be expanded to pythagorean fuzzy soft HX-groups and to apply more strategies for handling other hybrid models, such as m-polar soft HX-groups, bipolar soft HX-groups, and neutrosophic soft HX-groups. References [1] Lotfi A Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965. [2] Radwan Abu-Gdairi and Ibrahim Noaman. Generating fuzzy sets and fuzzy relations based on information. WSEAS Transactions on Mathematics, 20:178–185, 2021. [3] Michael Gr Voskoglou. Topological spaces on fuzzy structures. WSEAS Trans. Math, 21:624–628, 2022. [4] Asima Razzaque, Abdul Razaq, Ghaliah Alhamzi, Harish Garg, and Muham- mad Iftikhar Faraz. A detailed study of mathematical rings in q-rung orthopair fuzzy framework. Symmetry, 15(3):697, 2023. [5] K Atanassov. Intuitionistic fuzzy sets. fuzzy sets syst. 1986. [6] Krassimir T Atanassov. Intuitionistic fuzzy sets. Springer, 1999. [7] Abrar Hussain, Kifayat Ullah, Mohammed Nasser Alshahrani, Miin-Shen Yang, and Dragan Pamucar. Novel aczel–alsina operators for pythagorean fuzzy sets with ap- plication in multi-attribute decision making. Symmetry, 14(5):940, 2022. [8] Hanan Alolaiyan, Abdul Razaq, Humaira Ashfaq, Dilshad Alghazzawi, Umer Shuaib, and Jia-Bao Liu. Improving similarity measures for modeling real-world issues with interval-valued intuitionistic fuzzy sets. IEEE Access, 12:10482–10496, 2024. [9] Harish Garg, Dibakar Dutta, Palash Dutta, and Brindaban Gohain. An extended group decision-making algorithm with intuitionistic fuzzy set information distance measures and their applications. Computers & Industrial Engineering, 197:110537, 2024. [10] Ronald R Yager. Pythagorean fuzzy subsets. In 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), pages 57–61. IEEE, 2013. [11] Ghaliah Alhamzi, Saman Javaid, Umer Shuaib, Abdul Razaq, Harish Garg, and Asima Razzaque. Enhancing interval-valued pythagorean fuzzy decision-making through dombi-based aggregation operators. Symmetry, 15(3):765, 2023. [12] Hanan Alolaiyan, Umme Kalsoom, Umer Shuaib, Abdul Razaq, Abdul Wakil Baidar, A. Almuhaimeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5768 17 of 17 and Qin Xin. Precision measurement for effective pollution mitigation by evaluating air quality monitoring systems in linguistic pythagorean fuzzy dombi environment. Scientific Reports, 14(1):31944, 2024. [13] M Shazib Hameed, Salman Mukhtar, Haq Nawaz Khan, Shahbaz Ali, M Haris Ma- teen, and Muhammad Gulzar. Pythagorean fuzzy n-soft groups. Int. J. Electr. Com- put. Eng, 21:1030–1038, 2021. [14] Abdul Razaq and Ghaliah Alhamzi. On pythagorean fuzzy ideals of a classical ring. AIMS Math, 8(2):4280–4303, 2023. [15] Areej Almuhaimeed. Pythagorean fuzzy small submodules. European Journal of Pure and Applied Mathematics, 15(1):36–46, 2022. [16] FA Cotton. The intuitionistic fuzzy normal subgroup and its some equivalent propo- sitions, 1991. [17] Delaram Kahrobaei and Michael Anshel. Applications of group theory in cryptogra- phy. International Journal of pure and applied mathematics, 58:21–23, 2010. [18] Abdul Razaq, Shumaila Akhter, Awais Yousaf, Umer Shuaib, and Musheer Ahmad. A group theoretic construction of highly nonlinear substitution box and its applications in image encryption. Multimedia Tools and Applications, pages 1–22, 2022. [19] Azriel Rosenfeld. Fuzzy groups. Journal of mathematical analysis and applications, 35(3):512–517, 1971. [20] Li Hongxing. Hx group. BUSEFAL, pages 31–37, 1987. [21] M Hongahai and Z Wenyi. Direct product of hx-groups and hx-groups on direct product groups. Busefal, 54, 1993. [22] Rabah Kellil and Ferdaous Bouaziz. New investigations on hx-groups and soft groups. Italian Journal of Pure and Applied Mathematics on, (42), 2017. [23] Piergiulio Corsini. Hx-groups and hypergroups. Analele ştiinţifice ale Universităţii” Ovidius” Constanţa. Seria Matematică, 24(3):101–121, 2016. [24] Irina Cristea, Michal Novák, and Babatunde Oluwaseun Onasanya. Links between hx-groups and hypergroups. In Algebra Colloquium, volume 28, pages 441–452. World Scientific, 2021. [25] Piergiulio Corsini. Hypergroups associated with hx-groups. Analele ştiinţifice ale Universităţii” Ovidius” Constanţa. Seria Matematică, 25(2):49–64, 2017. [26] Li Hongxing. Hx group. Busefal, 33:31–37, 1987. [27] Luo Chengzhong. Fuzzy hx group. Busefal, 41-14:97–106, 1989. [28] Muthuraman. M. S Muthuraj. R, Manikandan. K. H and Sithar Selvam. P. M. Anti q-fuzzy hx group and its lower level sub hx groups. nternationalJournal of Computer Applications, 6:16–20, 2010.